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Unformatted text preview: is to ﬁnd polar representations of these points. In
10 Presumably, the name is derived from its resemblance to a stylized human heart. 810 Applications of Trigonometry
order for a point P to be on the graph of r = 2 sin(θ), it must have a representation P (r, θ)
which satisﬁes r = 2 sin(θ). If P is also on the graph of r = 2 − 2 sin(θ), then P has a (possibly
diﬀerent) representation P (r , θ ) which satisﬁes r = 2 sin(θ ). We ﬁrst try to see if we can
ﬁnd any points which have a single representation P (r, θ) that satisﬁes both r = 2 sin(θ) and
r = 2 − 2 sin(θ). Assuming such a pair (r, θ) exists, then equating11 the expressions for r
gives 2 sin(θ) = 2 − 2 sin(θ) or sin(θ) = 2 . From this, we get θ = π + 2πk or θ = 56 + 2πk
for integers k . Plugging θ = 6 into r = 2 sin(θ), we get r = 2 sin 6 = 2 2 = 1, which
is also the value we obtain when we substitute it into r = 2 − 2 sin(θ). Hence, 1, π is one
representation for the point of intersection in the ﬁr...
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